Group #31915689 — subgroups

Subgroups — #31915689

Order 18176.

486 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985803	C₂	1	4	yes	maximal
2	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
3	#22588504	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556062	C₂²	1	2	yes	—
7	C₂×#14556060	—	2	1	no	—
8	#17985865	—	2	1	no	—
9	#17985803	—	2	8	no	—
10	#17985775	—	2	2	no	—
11	C₂²×Q₁₆×C₇₁	—	2	1	no	—
12	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
14	#494×C₇₁	—	2	2	no	—
15	#494×C₇₁	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
21	#14556062	—	2	4	no	—
22	#14556060	—	4	2	no	—
23	C₇₁⋊#95	—	4	2	no	—
24	C₂²×Q₈×C₇₁	—	2	2	no	—
25	C₂×Q₁₆×C₇₁	—	2	4	no	—
26	C₂×Q₁₆×C₇₁	—	2	4	no	—
27	C₂×Q₁₆×C₇₁	—	2	2	no	—
28	C₂×Q₁₆×C₇₁	—	2	4	no	—
29	C₂×Q₁₆×C₇₁	—	4	2	no	—
30	C₂×Q₁₆×C₇₁	—	4	2	no	—
31	C₂×Q₁₆×C₇₁	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₇₁	—	2	4	no	—
35	C₄⋊Q₈×C₇₁	—	4	2	no	—
36	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	D₄	1	2	yes	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	—	4	2	no	—
42	C₄⋊Dic₇₁	—	4	2	no	—
43	C₇₁⋊C₁₆	—	2	4	no	—
44	C₂×Q₈×C₇₁	—	2	4	no	—
45	C₂×Q₈×C₇₁	—	4	2	no	—
46	C₂×Q₈×C₇₁	—	4	2	no	—
47	C₂×Q₈×C₇₁	—	4	2	no	—
48	C₂×Q₈×C₇₁	—	4	2	no	—
49	C₂×Q₈×C₇₁	—	4	2	no	—
50	C₂²×C₄×C₇₁	—	4	2	no	—
51	C₂²×C₄×C₇₁	D₈	1	2	yes	—
52	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₇₁	—	2	8	no	—
54	Q₁₆×C₇₁	—	4	8	no	—
55	Q₁₆×C₇₁	—	4	4	no	—
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₁	D₈	1	2	yes	—
58	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	2	yes	—
63	C₄⋊C₄×C₇₁	—	4	8	no	—
64	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₇₁	SD₁₆	1	2	yes	—
66	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₇₁	—	4	4	no	—
68	Q₈×C₇₁	—	4	4	no	—
69	Q₈×C₇₁	—	8	2	no	—
70	Q₈×C₇₁	—	8	2	no	—
71	C₂×C₄×C₇₁	—	2	4	no	—
72	C₂×C₄×C₇₁	—	4	4	no	—
73	C₂×C₄×C₇₁	—	8	2	no	—
74	C₂×C₄×C₇₁	—	8	2	no	—
75	C₂×C₄×C₇₁	#102	1	4	yes	—
76	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×Dic₇₁	—	8	2	no	—
83	C₂×Dic₇₁	—	8	4	no	—
84	C₈×C₇₁	#102	1	4	yes	—
85	C₈×C₇₁	C₄×D₄	1	2	yes	—
86	C₈×C₇₁	#121	1	2	yes	—
87	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
88	C₂²×C₇₁	#486	1	1	yes	characteristic
89	C₂²×C₇₁	#493	1	1	yes	characteristic
90	C₂²×C₇₁	#510	1	4	yes	—
91	C₄×C₇₁	—	2	4	no	—
92	C₄×C₇₁	—	8	4	no	—
93	C₄×C₇₁	#390	1	1	yes	characteristic
94	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
95	C₄×C₇₁	#486	1	1	yes	characteristic
96	C₄×C₇₁	#492	1	1	yes	characteristic
97	Dic₇₁	—	16	2	no	—
98	#13207	—	71	1	no	maximal, Sylow
99	C₂×C₇₁	#1957	1	1	yes	characteristic
100	C₂×C₇₁	#2141	1	2	yes	—
101	C₂×C₇₁	#2245	1	2	yes	—
102	C₂×C₇₁	#2251	1	2	yes	—
103	C₂×#494	D₇₁	1	1	yes	characteristic
104	C₂×#358	—	71	4	no	—
105	C₂×#346	—	71	1	no	—
106	#1846	—	71	1	no	—
107	C₇₁	#13207	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₁₆	—	2	1	no	—
109	C₂²×Q₁₆	C₂×D₇₁	1	2	yes	—
110	C₂×C₄⋊Q₈	—	2	1	no	—
111	C₂²×C₁₆	—	71	2	no	—
112	#494	—	2	2	no	—
113	#494	Dic₇₁	1	4	yes	complemented
114	C₂×#107	—	71	2	no	—
115	C₂×#106	—	142	1	no	—
116	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
117	#382	—	142	1	no	—
118	#358	—	142	8	no	—
119	#346	—	142	2	no	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂×Q₁₆	—	2	4	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	—	2	4	no	—
125	C₂×Q₁₆	—	4	2	no	—
126	C₂×Q₁₆	—	4	2	no	—
127	C₂×Q₁₆	C₂×Dic₇₁	1	4	yes	—
128	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	C₂×C₄⋊C₄	—	4	2	no	—
133	C₂×C₄⋊C₄	—	142	2	no	—
134	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
135	C₂×C₁₆	—	71	4	no	—
136	C₂×C₁₆	—	142	2	no	—
137	C₂×C₁₆	—	142	2	no	—
138	#107	—	142	4	no	—
139	#106	—	284	2	no	—
140	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
141	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
142	#95	—	284	2	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	284	2	no	—
151	C₂²×C₄	#11765850	1	2	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
153	Q₁₆	—	2	8	no	—
154	Q₁₆	—	4	8	no	—
155	Q₁₆	—	4	4	no	—
156	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
157	C₂×C₈	#11765850	1	2	yes	—
158	C₂×C₈	#11765853	1	2	yes	—
159	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
160	C₂×C₈	D₄×D₇₁	1	2	yes	—
161	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
162	C₂×C₈	#11765864	1	2	yes	—
163	C₄⋊C₄	—	4	8	no	—
164	C₄⋊C₄	—	284	2	no	—
165	C₄⋊C₄	—	284	2	no	—
166	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
167	C₄²	#11765853	1	2	yes	—
168	C₁₆	—	142	4	no	—
169	C₂³	C₂×#11765850	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	4	no	—
172	Q₈	—	8	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	568	2	no	—
179	C₂×C₄	—	568	4	no	—
180	C₂×C₄	#14556049	1	4	yes	—
181	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
182	C₂×C₄	#14556115	1	1	yes	characteristic
183	C₂×C₄	#14556124	1	1	yes	characteristic
184	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
185	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
186	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
187	C₈	#14556049	1	4	yes	—
188	C₈	D₄×Dic₇₁	1	2	yes	—
189	C₈	#14556099	1	2	yes	—
190	C₂²	C₂×#14556049	1	1	yes	characteristic
191	C₂²	#17986345	1	1	yes	characteristic
192	C₂²	#17986375	1	1	yes	characteristic
193	C₂²	#17986440	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	—	1136	2	no	—
197	C₄	#17985902	1	1	yes	characteristic
198	C₄	C₂×#14556049	1	1	yes	characteristic
199	C₄	#17986345	1	1	yes	characteristic
200	C₄	#17986371	1	1	yes	characteristic
201	C₂	#22589147	1	1	yes	characteristic
202	C₂	#22590060	1	2	yes	—
203	C₂	#22590585	1	2	yes	—
204	C₂	#22590609	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985803

C₂

yes

maximal

C₂×#17985775

C₂

yes

maximal, characteristic

#22588504

C₂

yes

maximal, characteristic

C₂×#494×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

C₂×#14556062

C₂²

yes

—

—

—

—

—

C₂²×Q₁₆×C₇₁

—

C₂²×Q₁₆×C₇₁

C₂²

yes

—

C₂×C₄⋊Q₈×C₇₁

—

#494×C₇₁

—

#494×C₇₁

C₄

yes

complemented

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556062

—

#14556060

—

C₇₁⋊#95

—

C₂²×Q₈×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

SD₁₆

yes

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#102

yes

—

C₈×C₇₁

C₄×D₄

yes

—

C₈×C₇₁

#121

yes

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

—

C₂×C₇₁

yes

100

C₂×C₇₁

#2141

yes

—

101

C₂×C₇₁

#2245

yes

—

102

C₂×C₇₁

#2251

yes

—

103

C₂×#494

D₇₁

yes

characteristic

104

C₂×#358

—

105

C₂×#346

—

106

#1846

—

107

C₇₁

#13207

yes

characteristic, complemented, Sylow

108

C₂²×Q₁₆

—

109

C₂²×Q₁₆

C₂×D₇₁

yes

—

110

C₂×C₄⋊Q₈

—

111

C₂²×C₁₆

—

112

#494

—

113

#494

Dic₇₁

yes

complemented

114

C₂×#107

—

115

C₂×#106

—

142

—

116

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

117

#382

—

142

—

118

#358

—

142

—

119

#346

—

142

—

120

C₂²×Q₈

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

C₂×Dic₇₁

yes

—

128

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

129

C₂²×C₈

C₂²×D₇₁

yes

characteristic

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

142

—

134

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

135

C₂×C₁₆

—

136

C₂×C₁₆

—

142

—

137

C₂×C₁₆

—

142

—

138

#107

—

142

—

139

#106

—

284

—

140

C₄×C₈

C₂×Dic₇₁

yes

—

141

C₄×C₈

C₂²⋊D₇₁

yes

—

142

#95

—

284

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

—

284

—

151

C₂²×C₄

#11765850

yes

—

152

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

153

Q₁₆

—

154

Q₁₆

—

155

Q₁₆

—

156

C₂×C₈

C₂²⋊Dic₇₁

yes

—

157

C₂×C₈

#11765850

yes

—

158

C₂×C₈

#11765853

yes

—

159

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

160

C₂×C₈

D₄×D₇₁

yes

—

161

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

162

C₂×C₈

#11765864

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

284

—

165

C₄⋊C₄

—

284

—

166

C₄²

C₂²⋊Dic₇₁

yes

—

167

C₄²

#11765853

yes

—

168

C₁₆

—

142

—

169

C₂³

C₂×#11765850

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

568

—

179

C₂×C₄

—

568

—

180

C₂×C₄

#14556049

yes

—

181

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

182

C₂×C₄

#14556115

yes

characteristic

183

C₂×C₄

#14556124

yes

characteristic

184

C₂×C₄

C₂×#11765850

yes

characteristic

185

C₂×C₄

C₂×#11765853

yes

characteristic

186

C₂×C₄

C₂×#11765853

yes

characteristic

187

C₈

#14556049

yes

—

188

C₈

D₄×Dic₇₁

yes

—

189

C₈

#14556099

yes

—

190

C₂²

C₂×#14556049

yes

characteristic

191

C₂²

#17986345

yes

characteristic

192

C₂²

#17986375

yes

characteristic

193

C₂²

#17986440

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

1136

—

197

C₄

#17985902

yes

characteristic

198

C₄

C₂×#14556049

yes

characteristic

199

C₄

#17986345

yes

characteristic

200

C₄

#17986371

yes

characteristic

201

C₂

#22589147

yes

characteristic

202

C₂

#22590060

yes

—

203

C₂

#22590585

yes

—

204

C₂

#22590609

yes

—

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#494×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#494	D₇₁
hypocenter	C₇₁	#13207
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#11765850

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#494×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
15. #494×C₇₁	C₄	semidirect
107. C₇₁	#13207	semidirect
113. #494	Dic₇₁	semidirect

H ≅ G/N

type

15. #494×C₇₁

C₄

semidirect

107. C₇₁

#13207

semidirect

113. #494

Dic₇₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#17985803	2
C₂×#17985775	2
#22588504	2
C₂×#494×C₇₁	2
#13207	71

subgroup

index

C₂×#494×C₇₁

Subgroups — #31915689

Conjugacy classes of subgroups

Complements

Maximal subgroups